How can the Universe be flat and isotropic?

If I understand these terms at all, I don't understand how the Universe can be Euclidean, isotropic and finite in spatial extent, all at the same time. It seems to me that if there's a finite amount of matter in it, then it needs to be closed to be isotropic and it can't be isotropic if it's flat... or if it's flat and isotropic, then there needs to be an infinite amount of stuff in it...

Is there something I'm misunderstanding here? How can these things be reconciled?
Thank you!

Well, isn't the theory that the very early Universe was small and it has expanded to the size it is now? Isn't the amount of energy in the Universe is finite? I thought the extent of space was finite too. If it's not, how could it have been small and expanding?

Anyway, if you want to know how our universe can be finite, perfectly spatially flat, and isotropic all at the same time, you only need to look at the old video game asteroids. This is what is known as a toroidal topology, and it is conceivable that our universe might well have this sort of topology (the overall topology of our universe is as yet unknown).

The stuff that spawned our observable universe was once very small. Doesn't mean the whole universe was, because we don't know how much lies beyond our observable part.

Okay but I also thought that the prevailing theory was that it was the beginning of space and time as well. Is this just a misunderstanding propagated by lay-people?

Anyway, if you want to know how our universe can be finite, perfectly spatially flat, and isotropic all at the same time, you only need to look at the old video game asteroids. This is what is known as a toroidal topology, and it is conceivable that our universe might well have this sort of topology (the overall topology of our universe is as yet unknown).

I thought being flat excluded the possibility of being closed. In order to be closed, you must be elliptic. Is this not the case?

So if we say that the Universe is isotropic, we're implying that it's closed?

Okay but I also thought that the prevailing theory was that it was the beginning of space and time as well. Is this just a misunderstanding propagated by lay-people?

Well, it's a misunderstanding that has unfortunately also been promoted by scientists with very high profiles. The correct statement is that we don't yet know how our universe began, or whether or not it started from some earlier space-time.

I thought being flat excluded the possibility of being closed. In order to be closed, you must be elliptic. Is this not the case?

Well, the first point is that we can't say it's absolutely flat. Rather, our universe is approximately flat. So obviously our universe is likely to be either a little bit open, or a little bit closed. We could never experimentally state that it's absolutely flat, because that would require infinite precision on the measurement. Furthermore our models of the very early universe that produce near flatness don't produce absolute flatness.

All that said, a torus is a flat topology that wraps back on itself. A fair way of visualizing this is the game asteroids, where if you go off the top of the screen, you appear on the bottom. If you go off the right of the screen, you appear on the left. So it wraps back on itself, and yet the screen is flat.

But no, isotropy and flatness are entirely different issues. The flatness has to do with how the average density relates to the rate of expansion. Isotropy just means that the universe looks the same in every direction.

I thought being flat excluded the possibility of being closed. In order to be closed, you must be elliptic. Is this not the case?

To expand on Chalnoth's answer:
When you hear someone talk about the universe, be aware of two things:
Most likely they will use the word for both the observable universe and the total universe.
Even if they get this right and are talking about the total universe, they don't mean the universe as it is, but the basic (mathematical) model. In the model, "flat" means spatially not closed, because that's the simplest configuration.
You could think of different models, like the torus Chalnoth mentioned, where a flat space is finite, but has a more complicated topology (not http://en.wikipedia.org/wiki/Simply_connected_space" [Broken]).

The real universe, however, could be something entirely different. We don't know how the unobservable universe looks like, for obvious reasons.

The flat torus is not "globally" isotropic. See for example, Jeffrey R. Weeks "The Shape of Space" or perhaps better still, page 15 of "Topology and the Cosmic Microwave
Background" at http://arxiv.org/PS_cache/gr-qc/pdf/0108/0108043v2.pdf" [Broken] by Janna Levin.

True, but that point is largely irrelevant. It is still locally isotropic. Only when you get to length scales large enough to wrap around the universe does the global isotropy start to become noticeable, and from our current observations we know that the wrap-around length, if there is one, is much larger than our observable universe.

True, but that point is largely irrelevant. It is still locally isotropic. Only when you get to length scales large enough to wrap around the universe does the global isotropy start to become noticeable, and from our current observations we know that the wrap-around length, if there is one, is much larger than our observable universe.

Do you by the way, know if the newest CMB data has anything to say about the topology? Have they successfully ruled out any of the many possibilities?

Do you by the way, know if the newest CMB data has anything to say about the topology? Have they successfully ruled out any of the many possibilities?

All that we can say so far is that there is no evidence of any sort of non-trivial topology. There probably won't ever be, unfortunately. It looks like the universe is enough larger than our observable region that we probably won't ever be able to say for sure.

Well, it's a misunderstanding that has unfortunately also been promoted by scientists with very high profiles. The correct statement is that we don't yet know how our universe began, or whether or not it started from some earlier space-time.

So the Big Bang is "the beginning" of the Universe only in the sense that it's the earliest discernible history of the known Universe? Does this mean "what came before the Big Bang" or "what caused the Big Bang" are sensible questions?

Well, the first point is that we can't say it's absolutely flat. Rather, our universe is approximately flat. So obviously our universe is likely to be either a little bit open, or a little bit closed. We could never experimentally state that it's absolutely flat, because that would require infinite precision on the measurement. Furthermore our models of the very early universe that produce near flatness don't produce absolute flatness.

How can any geometry be "a little bit open?" I'm pretty sure these are propositions: they're open or they're closed. It's like trying to say that something is a little bit infinite. Something can be big but it can't be a little infinite...

All that said, a torus is a flat topology that wraps back on itself. A fair way of visualizing this is the game asteroids, where if you go off the top of the screen, you appear on the bottom. If you go off the right of the screen, you appear on the left. So it wraps back on itself, and yet the screen is flat.

But no, isotropy and flatness are entirely different issues. The flatness has to do with how the average density relates to the rate of expansion. Isotropy just means that the universe looks the same in every direction.

If I understand this correctly, flatness refers to the curvature (or lack thereof) of space, which is independent of whether that space is closed or not...

Why do we think the Universe is isotropic?

I'm sorry but I'm a little confused on this point: do we think the Universe is closed?

So the Big Bang is "the beginning" of the Universe only in the sense that it's the earliest discernible history of the known Universe? Does this mean "what came before the Big Bang" or "what caused the Big Bang" are sensible questions?

Well, I should mention that not in all models are they both sensible questions. Stephen Hawking has a no boundary proposal, for instance, that still demands an answer to "what caused the big bang", but if true would mean that the question "what came before the big bang" doesn't make sense.

How can any geometry be "a little bit open?" I'm pretty sure these are propositions: they're open or they're closed. It's like trying to say that something is a little bit infinite. Something can be big but it can't be a little infinite...

It's about the radius of curvature. An open universe with strong curvature might have a curvature radius of the order of a Planck length, for instance. An open universe with very weak curvature, on the other hand, could have a curvature radius many times the size of the observable universe.

The cosmic microwave background is the same in every direction on the sky to one part in 100,000, once the dipole signal from our own motion is removed (before that removal, it's the same to one part in 10,000).

I'm sorry but I'm a little confused on this point: do we think the Universe is closed?

There is no evidence one way or the other at the current time. We just know that the radius of curvature is much larger than the observable universe.

However, I should mention that because the curvature radius must be larger than the observable universe with current observations, if we ever do detect a definitive curvature signal, we won't be able to say definitively that the entire universe has this same curvature: we could potentially be just in a part of the whole that happens to have open or closed curvature, and other parts might have different average curvatures.

To see how this sort of thing can happen, consider a torus in three dimensions. The interior surface of the torus has negative curvature, while the exterior surface has positive curvature.

The cosmic microwave background is the same in every direction on the sky to one part in 100,000, once the dipole signal from our own motion is removed (before that removal, it's the same to one part in 10,000).

How can any geometry be "a little bit open?" I'm pretty sure these are propositions: they're open or they're closed. It's like trying to say that something is a little bit infinite. Something can be big but it can't be a little infinite...

It's about the radius of curvature. An open universe with strong curvature might have a curvature radius of the order of a Planck length, for instance. An open universe with very weak curvature, on the other hand, could have a curvature radius many times the size of the observable universe.

So you're trying to describe a geometry with weaker curvature as being "more open?" Okay but I don't think that's any better a characterization than describing a large number as being more infinite...

I think this goes back to my original post but, this time, I'm not conflating flatness with openness. If we think the Universe is isotropic then don't we think it's closed? Conversely, how can it be isotropic if it's open?

Now that I'm thinking about it, the only way I can reconcile isotropy with openness is if an open Universe implies an infinite amount of energy in it (to fill the infinite amount of space). Otherwise, a closed Universe can only have a finite amount of energy but will be isotropic. Is this right?

Well, it's a misunderstanding that has unfortunately also been promoted by scientists with very high profiles. The correct statement is that we don't yet know how our universe began, or whether or not it started from some earlier space-time.

If the Big Bang theory doesn't specify whether the Big Bang is the beginning of the Universe then what are people talking about when they say that we can reckon the Universe back to the first Planck time after the beginning? I understand that this isn't necessarily the beginning but what is it? Is it a point in time that the equations say that the Universe was a singularity?

Its quite interesting, pertaining to this subject. My personal belief, if I had to say, would be that the universe is infinite in the sense that we cannot define edges, and that we have no knowledge of things outside the supposed boundaries of the universe. However, I do not mean infinite in the sense that we can continue to travel in a particular direction forever, but rather, as a friend of mine postulated, that infinite is a "state," rather than a mathematical number or measurement. We have measured the amount of energy that comes to the Earth from the rest of the universe, and we have found that the same amount of energy approaches the earth from all directions! Of course this would imply that the earth is the center of the universe, but that, of course, is a quite naive way to view the universe! So, in my eyes, we, the earth, are either in the center of the universe or there is energy coming from places that we cannot see (aka the "infiniteness" of the universe).

The universe is obviously not flat, because we live in a 3 dimensional space where we can travel in any direction. However, maybe if you consider more dimensions one would eventually reach a space where it would appear flat, but we know that will not occur in the 4th dimension, as postulated by Einstein....

However, I do not mean infinite in the sense that we can continue to travel in a particular direction forever,

Well, there is really no question that we could do this, had we the technology. The only problem is that because of the accelerated expansion, we can't ever reach stuff that is currently around 10 billion light years away or so. This doesn't stop us from continuing in that direction, but the stuff that is currently around that far away will always be moving away faster than we can get there.

The universe is obviously not flat, because we live in a 3 dimensional space where we can travel in any direction. However, maybe if you consider more dimensions one would eventually reach a space where it would appear flat, but we know that will not occur in the 4th dimension, as postulated by Einstein....

Uh, what? This has nothing to do with flatness.

As I said before, flatness is about the relationship between the rate of expansion and the energy density of the universe. If things are moving too rapidly away from one another, the universe is "open". If things aren't moving that rapidly away, then the universe is "closed".

Isotropic just means "same in every direction". Perhaps you meant homogeneous?

I thought isotropy meant everything looked the same (regardless of which direction you looked) no matter where you were in the Universe. However, I see that your use of the word is what is meant in the context of cosmology...

Not at all. The simplest flat and open universes are entirely isotropic (and homogeneous). These properties are entirely orthogonal to one another.

You didn't address the rest of my post. I suspect that the source of any misunderstanding I'm having here can be determined from that portion of the post...

A flat and open universe need not be isotropic. If we were at the edge of such a universe with finite mass, we will not observe isotropy, right?

If the Universe were closed and homogeneous then it will be isotropic. Therefore, either the Universe is open and we're at its center or there's an infinite amount of mass (so there is no center), or it's closed (so there's still no center). Doesn't this follow?

Closed is a statement about curvature. Whether or not the universe wraps back on itself is a statement about topology. It is perfectly possible to have a flat universe with a topology that wraps back on itself. Just as it is, in principle, possible to have a closed universe that doesn't extend far enough to wrap back on itself.

When I was using the term "closed", I was making a statement of topology. That's why I was linking to my use of the term in this important post of mine. I've heard of closed curves but I've never seen this word used to describe curvature before...

Looking back on my past posts, do my questions make more sense to you now?